Changes between Version 5 and Version 6 of u/erica/scratch3
- Timestamp:
- 04/03/16 14:11:46 (9 years ago)
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u/erica/scratch3
v5 v6 30 30 == Optically thick limit == 31 31 32 And we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the optical depth [[latex($\tau=1/\kappa$)]], to the radiation's scale height, [[latex($L^{-1}=\frac{\nabla E}{E}$)]]. 32 From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the optical depth [[latex($\tau=1/\kappa$)]], to the radiation's scale height, [[latex($L^{-1}=\frac{\nabla E}{E}$)]]. Thus, as [[latex($R\rightarrow 0$)]], we have: 33 33 34 Thus, as [[latex($R\rightarrow 0$)]], we have that [[latex($\frac{\tau}{L}\rightarrow 0$)]] - i.e. radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime. 34 [[latex($\boxed{\frac{\tau}{L}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]] 35 36 That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime. 35 37 36 38 The rad diffusion equation then becomes, … … 38 40 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)$)]] 39 41 40 which is equation 6.59 in Drake's book.42 which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer. 41 43 42 44 == Free streaming limit == 45 46 In the other limit, [[latex($R\rightarrow \infty$)]], we have: 47 48 [[latex($\boxed{\frac{\tau}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]] 49 50 That is, a photon travels an infinite distance before it interacts with another particle -- i.e. we are in the free-streaming limit.